Peirce identifies inference with a process he describes as symbolization. Let us consider what that might imply.

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information. (467).

Even if it were only a rough analogy between inference and symbolization, a principle of logical continuity, what is known in physics as a correspondence principle, would suggest parallels between steps of reasoning in the neighborhood of exact inferences and signs in the vicinity of genuine symbols. This would lead us to expect a correspondence between degrees of inference and degrees of symbolization extending from exact to approximate (non-demonstrative) inferences and from genuine to approximate (degenerate) symbols.

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

In the first place there are likenesses or copies — such as statues, pictures, emblems, hieroglyphics, and the like. Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.

The second kind of representations are such as are set up by a convention of men or a decree of God. Such are tallies, proper names, &c. The peculiarity of these conventional signs is that they represent no character of their objects.

The third and last kind of representations are symbols or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all words and all conceptions. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (467–468).

In addition to Aristotle, the influence of Kant on Peirce is very strongly marked in these earliest expositions. The invocations of “conceptions of the understanding”, the “use of concepts” and thus of symbols in reducing the manifold of extension, and the not so subtle hint of the synthetic à priori in Peirce’s discussion, not only of natural kinds but also of the kinds of signs leading up to genuine symbols, can all be recognized as pervasive Kantian themes.

In order to draw out these themes and see how Peirce was led to develop their leading ideas, let us bring together our previous Figures, abstracting from their concrete details, and see if we can figure out what is going on.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework we used before.

Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous; we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous. (468–469).

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol. But suppose we know of no symbol for neat, swine, sheep, and deer except cloven-hoofed animals. (469).

This is apparently a stock example of inductive reasoning which Peirce is borrowing from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores.

In view of the analogical symmetries the disjunctive term shares with the conjunctive case, we can run through this example in fairly short order. We have an aggregate of four terms:

Suppose is the logical disjunction of the above four terms:

Figure 2 diagrams the situation before us.

Figure 2. Disjunctive Term u, Taken as Subject

Here we have a situation that is dual to the structure of the conjunctive example. There is a gap between the logical disjunction in lattice terminology, the least upper bound (lub) of the disjoined terms, and what we might regard as the natural disjunction or natural lub of these terms, namely, cloven-hoofed.

Once again, the sheer implausibility of imagining the disjunctive term would ever be embedded exactly as such in a lattice of natural kinds leads to the evident naturalness of the induction to namely, the rule that cloven-hoofed animals are herbivorous.

At this point in his inventory of scientific reasoning, Peirce is relating the nature of inference, information, and inquiry to the character of the signs mediating the process in question, a process he is presently describing as symbolization.

In the interest of clarity let’s draw from Peirce’s account a couple of quick sketches, designed to show how the examples he gives of conjunctive terms and disjunctive terms might look if they were cast within a lattice-theoretic frame.

Let’s examine Peirce’s example of a conjunctive term — spherical, bright, fragrant, juicy, tropical fruit — within a lattice framework. We have these six terms:

If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once.

In other words, if something is said to be then we may guess fairly surely is really an orange, in short, has all the additional features otherwise summed up quite succinctly in the much more constrained term where means an orange.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

What Peirce is saying about not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction in lattice terms, the greatest lower bound (glb) of the conjoined terms, and what we might regard as the natural conjunction or natural glb of these terms, namely, an orange. That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose. The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between and

As I suggested in an earlier discussion, the difference that makes a difference in Peirce’s contribution to our understanding of inference and reference is “due to his concurrent development of the logic of relative terms and the mathematics of relations, especially triadic relations.” It is only with the addition of these tools to our toolbox that we begin to form models adequate to the complexity of the object phenomena, namely, the whole panoply of activities involved in observation, conceptualization, communication, and inquiry.

I’ll be bringing these tools to bear as needed in the current engagement with Peirce’s incipient information theory, but only as the application calls for them. For a more general grounding in the relational logic and mathematics Peirce was developing in parallel at this time, see my notes, still in progress, on his 1870 Logic of Relatives:

Selection 1 opens with Peirce proposing, “The information of a term is the measure of its superfluous comprehension”, and it closes with his offering the following promise:

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information.

Summing up his account to this point, Peirce appears confident he’s kept his promise. Promising on our own account to give it another pass, we’ll let him have the last word — for now:

We have now seen how the mind is forced by the very nature of inference itself to make use of induction and hypothesis.

But the question arises how these conclusions come to receive their justification by the event. Why are most inductions and hypotheses true? I reply that they are not true. On the contrary, experience shows that of the most rigid and careful inductions and hypotheses only an infinitesimal proportion are never found to be in any respect false.

And yet it is a fact that all careful inductions are nearly true and all well-grounded hypotheses resemble the truth; why is that? If we put our hand in a bag of beans the sample we take out has perhaps not quite but about the same proportion of the different colours as the whole bag. Why is that?

The answer is that which I gave a week ago. Namely, that there is a certain vague tendency for the whole to be like any of its parts taken at random because it is composed of its parts. And, therefore, there must be some slight preponderance of true over false scientific inferences. Now the falsity in conclusions is eliminated and neutralized by opposing falsity while the slight tendency to the truth is always one way and is accumulated by experience. The same principle of balancing of errors holds alike in observation and in reasoning.

Peirce now turns to his example of a conjunctive term, which he uses to show the connection between iconic reference and abductive inference.

A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term:

spherical, bright, fragrant, juicy, tropical fruit.

Such a term, formed by the sum of the comprehensions of several terms, is called a conjunctive term. A conjunctive term has no extension adequate to its comprehension. Thus the only spherical bright fragrant juicy tropical fruit we know is the orange and that has many other characters besides these. Hence, such a term is of no use whatever. If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once. On the other hand, if the conjunctive term is subject and we know that every spherical bright fragrant juicy tropical fruit necessarily has certain properties, it must be that we know more than that and can simplify the subject. Thus a conjunctive term may always be replaced by a simple one.

So if we find that light is capable of producing certain phenomena which could only be enumerated by a long conjunction of terms, we may be sure that this compound predicate may be replaced by a simple one. And if only one simple one is known in which the conjunctive term is contained, this must be provisionally adopted.

When I was first learning information theory, one of my favorite tag lines was:

Redundancy is the Essence of Information

It’s a bit tongue-in-cheek but none-the-less apt.

Although I had happened on hints of Peirce’s information theory in the microfilm edition of his Nachlass, only when the Chronological Edition of his Writings started coming out did I begin to grasp the full scope of what he was setting forth in his early Lectures on the Logic of Science.

And when I eventually lit on Peirce’s otherwise enigmatic “superfluous comprehension” I could hardly help but flash back to my earlier kenning of redundancy and sense the two phrases must be pointing to the same critical property of information. However things may turn out, I think this connection gives us a clue worth following up.